Cook the meatballs according to the recipe directions.
Meanwhile, cook the pasta according to the package directions. Reserve ¼ cup of the cooking water; drain the pasta and return it to the pot.
While the meatballs and pasta are cooking, heat the butter in a large skillet over medium heat. Add the mushrooms and thyme and cook until tender, 8 to 10 minutes.
Toss the mushroom mixture with the pasta, meatballs, spinach, pecorino, and the reserved cooking water. Season with ½ teaspoon salt and ¼ teaspoon pepper.